3,383 research outputs found
Spatially independent martingales, intersections, and applications
We define a class of random measures, spatially independent martingales,
which we view as a natural generalisation of the canonical random discrete set,
and which includes as special cases many variants of fractal percolation and
Poissonian cut-outs. We pair the random measures with deterministic families of
parametrised measures , and show that under some natural
checkable conditions, a.s. the total measure of the intersections is H\"older
continuous as a function of . This continuity phenomenon turns out to
underpin a large amount of geometric information about these measures, allowing
us to unify and substantially generalize a large number of existing results on
the geometry of random Cantor sets and measures, as well as obtaining many new
ones. Among other things, for large classes of random fractals we establish (a)
very strong versions of the Marstrand-Mattila projection and slicing results,
as well as dimension conservation, (b) slicing results with respect to
algebraic curves and self-similar sets, (c) smoothness of convolutions of
measures, including self-convolutions, and nonempty interior for sumsets, (d)
rapid Fourier decay. Among other applications, we obtain an answer to a
question of I. {\L}aba in connection to the restriction problem for fractal
measures.Comment: 96 pages, 5 figures. v4: The definition of the metric changed in
Section 8. Polishing notation and other small changes. All main results
unchange
Euler systems for Rankin--Selberg convolutions of modular forms
We construct an Euler system in the cohomology of the tensor product of the
Galois representations attached to two modular forms, using elements in the
higher Chow groups of products of modular curves. We use this Euler system to
prove a finiteness theorem for the strict Selmer group of the Galois
representation when the associated p-adic Rankin--Selberg L-function is
non-vanishing at s = 1.Comment: Revised version with many updates and correction
A note on p-adic Rankin--Selberg L-functions
We prove an interpolation formula for the values of certain -adic
Rankin--Selberg -functions associated to non-ordinary modular forms.Comment: Updated version, with minor corrections. To appear in Canad. Math.
Bulleti
The Marr Conjecture and Uniqueness of Wavelet Transforms
The inverse question of identifying a function from the nodes (zeroes) of its
wavelet transform arises in a number of fields. These include whether the nodes
of a heat or hypoelliptic equation solution determine its initial conditions,
and in mathematical vision theory the Marr conjecture, on whether an image is
mathematically determined by its edge information. We prove a general version
of this conjecture by reducing it to the moment problem, using a basis dual to
the Taylor monomial basis on .Comment: 52 pages, 4 figure
The anisotropic multichannel spin- Kondo model: Calculation of scales from a novel exact solution
A novel exact solution of the multichannel spin- Kondo model is presented,
based on a lattice path integral approach of the single channel spin-1/2 case.
The spin exchange between the localized moment and the host is of -type,
including the isotropic limit. The free energy is given by a finite set
of non-linear integral equations, which allow for an accurate determination of
high- and low-temperature scales.Comment: 18 pages, 7 figure
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